Figure REF  shows the performance of our proximal Jacobi scheme and the two-level algorithm.
All residuals correspond to infinity norms; in our scheme, the plotted primal and dual residuals are \(\Vert Ax^k - b\Vert _{\infty }\)  and \(\Vert d^k\Vert _{\infty }\) ; in the two-level scheme, they are \(\max \lbrace \Vert p^{g,k} - \bar{p}^{g,k}\Vert _{\infty }, \Vert \hat{p}^{g,k} - \bar{p}^{g,k}\Vert _{\infty }\rbrace \)  and the infinity norm of [1]} (plotted as a function of the number of cumulative iterations), respectively.
We observe that for the first few iterations (up to 25) and for small target tolerances (up to \(10^{-3}\) ), the two algorithms have similar behavior in terms of reducing the optimality errors.
However, our proposed scheme is able to decrease the residuals at an almost linear rate even for tighter tolerances, whereas the decrease for the two-level scheme seems to become worse than linear. This is likely because Algorithm REF  allows \(\rho \)  to decrease; indeed, we observe in Figure REF  that the primal and dual residuals are roughly within a factor of \(\chi = 10\)  of each other at all iterations.
Finally, note that the run times of both schemes are expected to be similar, since the per-iteration cost in either scheme is dominated by the solution of (single-period) ACOPF NLP problems.
The actual run time is a function of the number of parallel processes available which we study in the next section.
<FIGURE>